Reductions related with Hopf maps

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by  Vahagn Yeghikyan

Reductions related with Hopf maps
Vahagn Yeghikyan
Yerevan State University, 1 Alex Manoogian St., Yerevan, 0025, Armenia
1 Abstract
1
We consider the reductions of 2p-dimensional particle system (p = 2,4,8),
0
2 associated withtheHopfmap. For thethirdHopfmapweexplicitly construct
n the functions associated to the symmetry related to the rotations in the fiber.
a
J

2 1 Introduction
2

] It is known that the systems describing motion of particle in the field of of Dirac
h
p an Yang monopoles can be constructed using the reduction procedure associated
- with the first and second Hopf maps [1]. The Hopf maps(fibrations) are fibra-
h
t tions of spheres over spheres with the fiber-sphere [2]. There are four Hopf maps:
a
m S2n−1/Sn−1 = Sn, (n = 1,2,4,8). The Dirac and Yang monopoles are related with
[ the first and the second ones respectively. Zero Hopf map is related with anyons

1 (or magnetic vortices)[3], while for the last Hopf map the mentioned procedure does
v not exist. Moreover, it is even unclear which sort of monopole should arise after the
9
reduction. The problem comes from the fact that the algebra of the octonions is
9
2 not associative (equivalently the fiber of the last fibration is not a group manifold).
4
Therefore, the transformation, which leave invariant the coordinates of base of the
.
1
third Hopf fibration are not isometries of bundle space.
0
1 The goal of current paper is to investigate the problems arising while trying to
1
construct the reduction procedure related to the third Hopf map. For this purpose
:
v we formulate the reduction procedures associated with the Hopf maps ([1, 4]) in
i
X terms of real coordinates and Clifford algebras.
r Together with geometric methods (for review, see e.g. [5]) the algebraic un-
a
derstanding of the nature of the Hopf maps leaves to no surprise that important
differences are encountered between the Hopf maps.
The paper is arranged as follows.
In the Second Section we present an explicit description of the Hopf maps in
terms needed for our purposes.
In the Third section we employ the Hopf maps to reduce the bosonic free-particle
systems to lower dimensional systems with magnetic SU(2) monopoles.
2 Hopf maps
The Hopf maps (or Hopf fibrations) are the fibrations of the sphere over a sphere,
S2p−1/Sp−1 = Sp, p = 1,2,4,8. These fibrations reflect the existence of normed
division algebras: real (R, p = 1), complex (C, p = 2), quaternionic (H, p = 4) and
octonionic (O, p = 8) numbers.
2.1 Normed Division algebras
Any element of normed division algebras can be expressed via the generating ele-
ments of the algebra e .
µ
x = xn +xµe , µ = 1,...,n 1, n = 1,2,4,8, (2.1)
µ
−
where the generating elements satisfy the following multiplicative rule:
e e = δ +C e , (2.2)
µ ν µν µνλ λ
−
where C are constants antisymmetric under any permutations of indices. Here
µνλ
and in the further we will use bold style to denote the elements of algebra and
normal style for real elements.
The conjugationandnormaredefined byanalogywith complex numbers(n = 2):
x¯ = x x e , x √xx¯ = √x x .
n µ µ a a
− | | ≡
The greek symbols µ,ν,λ run 1,...,n 1, while the latin symbols a,b,c = 1,...,n.
−
It was proven[6], that one can construct the constants C so that the algebras
µνλ
have division operation only for the dimensions n = 1,2,4,8. It is clear that for
real and complex numbers we have C = 0. For quaternionic numbers we define
µνλ
C = ε , where ε are the elements of totally antisymmetric tensor and for
µνλ µνλ µνλ
the octonions we have
C = C = C = C = C = C = C = 1, (2.3)
123 147 165 246 257 354 367
while all other non-vanishing components aredetermined by thetotal antisymmetry.
Eachtimewepassfromalower dimensional algebratothenextone, welosesome
symmetry. Hence, first we lose the fact that every element is its own conjugate, then
we lose commutativity, then we lose associativity [7]. However, the last- octonionic
algebra has a weaker property called alternativity which implies any subalgebra
consisting of two elements is associative(for associativity we should have three). For
more information about modern status of the theory of normed division algebras see
an excellent review [7].
One can consider the elements (2.1) as columns with n real elements. Using (2.2)
one can write down the multiplicative rule for this columns:
(xy) = x γc yb, (γc) = δ δc +δcδ +δcδ Cµ (γc) , (2.4)
a c ab ab − an b a bn n ab − ab ≡ ab
where we have chosen C = 0 if at least one index is equal to n. Since we deal
abc
with Euclidean space, there is no difference between upper and lower indices. Here
and further we will denote the columns by the normal letters without indices.
One can see, that from requirement xy = x y ( x,y R,C,H,O) and the
| | | || | ∀ ∈
definition (2.4) it follows, that
γµ,γν = δµν, γn = 1n×n, (γµ)T = γµ, (2.5)
{ } − −
where .,. denotes anticommutator and T-transpose of matrices(See e.g. [8]). This
{ }
is all we need to know about the normed division algebras. Now, let us pass to the
description of Hopf maps.
2
2.2 Hopf maps (Normed division algebras)
Let us describe the Hopf maps using normed division algebras. For this purpose, we
consider the functions x(u ,u¯ ),x (u ,u¯ )
α α p+1 α α
x = 2u¯ u , xn+1 = u¯ u u¯ u , (2.6)
1 2 1 1 2 2
−
where u ,u are complex numbers for n = 2 case (first Hopf map), quaternionic
1 2
numbers for the n = 4 case (second Hopf map) and octnionic numbers for n = 8
(third Hopf map) (see,e.g. [5]). One can consider them as coordinates of the 2n-
dimensional space R2n (n = 2 for u complex numbers; n = 4 for u quaternionic
1,2 1,2
numbers, n = 8 for u octonionic numbers). In all cases x is a real number
1,2 n+1
while x is, respectively, complex number (n = 2), a quaternionic one (n = 4), and
octonionic one (n = 8)
x xn +e xµ. (2.7)
µ
≡
One could immediately check that the following equation holds:
r2 x¯x+(xn+1)2 = (u¯ u +u¯ u )2 R4. (2.8)
1 1 2 2
≡ ≡
Thus, defining the (2n 1)-dimensional sphere in
−
mathbbR2n of radius R, u¯ u = R2, we will get the p-dimensional sphere in Rn+1
α α
with radius r = R2.
The expressions (2.6) can be easily inverted by the use of equality (2.8)
u = gr , (2.9)
α α
where
r +xn+1 x
r = , r r = , g¯g = 1.
1 r 2 2 ≡ + 2(r+xn+1)
p
It follows from the last equation in (2.9) that g parameterizes the (n 1)-
−
dimensional sphere of unit radius.
Using above equations, it is easy to describe the first three Hopf maps. Indeed,
for n = 1,2,4 the functions x,x remain invariant under the transformations
n+1
u Gu , where G¯G = 1 (2.10)
α α
→
Therefore, G parameterizes the spheres Sn−1 of unit radius. Taking into account
the isomorphism between these spheres and the groups,for n = 1,2,4:S0 = Z ,
2
S1 = U(1), S3 = SU(2), we get that (2.6) is invariant under G group transforma-
−
tions for n = 1,2,4 (where G = Z for n = 1, G = U(1) for n = 2, and G = SU(2)
2
for n = 4).
For the octonionic case n = 8 situation is more complicated. Because of losing
associativity the standard transformation u that leaves invariant coordinates x,x
α 9
will not be just (2.10). Instead, we should write
3
Its modification can be easily obtained using (2.9):
(Gu )(u¯ u )
1 1 α
u (Gg)(g¯u ) = . (2.11)
α α
7→ u¯ u
1 1
Also, for the n = 8 case the bundle S7 is not isomorphic with any group. So, one
can expect further troubles in the extensions of the constructions, related with the
lower Hopf maps to the third one.
2.3 Hopf maps (Spinor representation)
One can consider a 2n-dimensional column U consisting of the real coordinates u ,
α,a
α = 1,2, a = 1,...,n:
U = (u ,...,u ,u ,...,u ).
1,1 1,n 2,1 2,n
Using this denotation one can rewrite (2.6) in the following form:
xA = UΓAU, A = 1,...,n+1 (2.12)
where
Γµ = 0 λµ , Γn = 0 1n×n , Γn+1 = −1n×n 0 ,
(cid:18) λµ 0 (cid:19) (cid:18) 1n×n 0 (cid:19) (cid:18) 0 1n×n (cid:19)
−
(2.13)
where
(λµ) = δ δµ +δµδ +C (2.14)
ab − an b a bn µab
(compare with (2.4)). This matrices satisfy the relations (2.5). One can check, that
thematricesΓA aretheEuclideangamma-matricessatisfyingtheanticommutational
relations:
ΓA,ΓB = δAB12n×2n. (2.15)
In [9] all the three Hopf m(cid:8)aps we(cid:9)re explicitly constructed using spinor repre-
sentations of SO(1,n + 1). For the complex and quaternionic (n = 2,4) case it
was shown the direct connection between this description and the one using normed
division algebras.
It is obvious, that for n = 2,4 we have reducible representation of Clifford
algebra(wehaven = 2,4generatingelements and4and8dimensionalrepresentation
respectively).
n = 8 is the only case for which the constructed matrices form an irreducible
representation of Clifford algebra. E.g. nine matrices ΓA have dimension 16 = 2[9/2].
It is clear, that in this representation all the matrices ΓA are symmetric. These is, in
fact, such representation, where the matrix of charge conjugation is identity matrix:
(CT)−1ΓAC = (ΓA)T = ΓA, C = 116×16
In [9] it was shown, that the infinitesimal transformation (2.11) can be presented
in the following form:
1
δU = ω (UTΓABCDU)ΓCDU (2.16)
AB
−6
4
3 Hopf maps and reductions
LetusapplytheobtainedformulaefortheHopffibrationstoreducethe2n-dimensional
free particle system to a lower dimensional one. For this reason we consider La-
grangian in terms of the coordinates of fiber and base of Hopf fibrations:
g(r )
= α ¯r˙ r˙ +2Re (¯r˙ g¯)(g˙r ) +r¯g˙g˙ = (3.17)
2n α α α α
L 2
(cid:0) (cid:0) (cid:1) (cid:1)
g
= r˙ ¯r˙ +2rv A v˙ +rv˙ v˙ , a,b,c,d = 1,...,8
α α a ab b a a
2
(cid:0) (cid:1)
where
x (Σcd) x˙ [λµ,λν]
A = c ab d, Σµν = , Σµn = Σnµ = λν, µ,ν = 1,...,7
ab
2r(r+x ) 2 −
9
(3.18)
is precisely the potential of U(1) Dirac, SU(2) Yang and S0(8) monopoles [10] with
λ be the two, four or eight-dimensional gamma-matrices (for n = 2,4,8 respec-
i
tively). Thefunctionsv aretheEuclideancoordinatesofthefiberofHopffibrations:
i
S1 for the first, S3 for the second and S7 for the third Hopf map. g = v + e v ,
8 a a
v v = 1 and express via projective coordinates of S2n−1 as follows:
a a
1 y2 2y n−1
v = − , v = µ , y2 = y2 (3.19)
n 1+y2 µ 1+y2 µ
Xµ=1
Taking into account the last expressions, we can represent the Lagrangian in the
following form:
g y˙ y˙
= ¯r˙ r˙ +2rD y˙ +2r µ µ ,
L 2 (cid:18) α α µ µ (1+y2)2(cid:19)
where
1
D = A (1 y2)+y A +2y A y , n = 2,4,8. (3.20)
µ (1+y2)2 nµ − ν νµ ν nν µ
(cid:0) (cid:1)
Let us replace our Lagrangian by the variationally equivalent one, performing
Legendre transformation of the “isospin” varyables y .
µ
After some work we find
g (p 2grD )2
= p y˙ + ¯r˙ r˙ (1+y2)2 µ − µ (3.21)
int µ µ α α
L 2 − 8rg
The generator of transformations (2.11) is defined, in these terms, as follows
1 p 1
Iµ = (1+y2)Sµ ν, where Sµ = 2y y +(1 y2)δµ +2y Cµ , (3.22)
2 ν 2 ν 1+y2 µ ν − ν λ νλ
(cid:0) (cid:1)
with C be structure constants of complex, quaternionic and octonionic algebra.
µνλ
5
Remark. It is obvious, that for the case of associative algebras(complex and
quaternionic) the transformation (2.11) form the symmetry of the initial Lagrangian
(3.17). However, for the case of octonions the lack of associativity leads to the
fact, that this transformations do not preserve the Lagrangian and, therefore, the
quantities I are not the integrals of motion of the system. We will discuss this
µ
below.
Taking into account the equalities
IµIµ p2 x˙ x˙
SST = 1n−1, = (1+y2)2 r˙Ar˙A rDµDµ(1+y2)2 = A A (3.23)
2gr 16rg − 4r
we can represent the Lagrangian in very transparent form
x˙ x˙ 1I I
A A µ µ
= g +p y˙ +rgJ A , (3.24)
int µ µ ab ab
L 8r − 4 2gr
where
1 y2
J = y p y p , J = J = − p +(y p )y , n = 2,4,8. (3.25)
µν µ ν ν µ µn nµ µ ν ν µ
− − 2
are the generators of SO(n) rotations.
3.1 n = 2 complex case
Inthis case we have a,b = 1,2and, therefore, oneelement J : J = J = p. It is
ab 12 21
−
easy to check, that this element is a constant of motion of the system and therefore
we can fix its value to be equal to a constant s. The term with y˙ disappears because
it becomes full time derivative and finally we find the reduced Lagrangian:
x˙ x˙ s2
A A
= g +rsA , A = 1,2,3, (3.26)
3 D
L 8r − 2gr
where A is the vector-potential of Dirac monopole.
D
3.2 n = 4 quatrenionic case
We have already mentioned, that for n = 4 the representation of Clifford algebra is
not minimal and, therefore not all the components of A are independent. Using
ab
the properties of ε one can find the following connection between this elements:
µνλ
ε A = 2A (3.27)
λµν µν λn
And, therefore, we find
J A = P A˜ (3.28)
ab ab µ µ
where
1 1
˜
A = ε A , P = J ε J . (3.29)
λ λµν µν λ nλ λµν µν
2 − 2
6
Let us mention that the following identity obeys:
1
I = J ε J (3.30)
µ nλ λµν µν
− − 2
Using this denotations, one can rewrite the Lagrangian (3.24) as follows:
x˙ x˙ 1I I
L = g A A +p y˙ 4rgP A˜ µ µ (3.31)
8 µ µ µ µ
8r − − 4 2gr
The quantities P together with I form so(4) = so(3) so(3) algebra of symmetries
µ µ
×
of S3. E.g. they obey the following commutation relations:
P ,I = 0, P ,P = ε P , I ,I = ε I , I I = P P (3.32)
µ ν µ ν µνλ λ µ ν µνλ λ µ µ µ µ
{ } { } { }
Now, we are ready to fix the values of the integrals of motion and hence, to
perform the reduction. Without loss of generality we can fix
I = I = 0, I = s (3.33)
1 2 3
Because of the relations (3.32) we can denote:
z¯ z
P+ = P2 +ıP1 = ıs ısh−, P− = P¯+ = ıs ısh+ (3.34)
− 1+zz¯ ≡ − 1+zz¯ ≡
1 zz¯
P = s − sh , z,z¯ = (1+zz¯)2
3 3
− 1+zz¯ ≡ − { }
and the Lagrangian (3.31) will take the following form:
gx˙ x˙ z¯z˙ zz¯˙ s2 g
A A
= is − sh (z,z¯)A , g , µ = 1,...,5,
Lred 2 − 1+zz¯ − µ µ − 2r2g ≡ 2r
e
(3.35)
e
where the quantities h±,h3 are defined by (3.34).e
The second term in the above reduced Hamiltonian is the one-form defining the
symplectic (and Ka¨hler) structure on S2, while h given in (3.34) are the Killing
k
potentials defining the isometries of the Ka¨hler structure. We have in this way
obtained the Lagrangian describing the motion of a five-dimensional isospin particle
in the field of an SU(2) Yang monopole. The metric of the configuration space is
defined by the expressions g = g δ . For a detailed description of the dynamics
µν 2r µν
of the isospin particle we refer to [11].
e
3.3 n = 8 octonionic case
It was already mentioned that because of lack of associativity of the octonionic alge-
bra the transformations (2.11) and therefore the functions I do not form isometries
µ
of the Lagrangian (3.17).
7
It is seen from (3.23), that I I = J J defines constant of motion of the system,
a a ij ij
incomplete analogywith the lower Hopf map. Reducing thesystem by this constant
of motion, we shall get the system with 30(= 2 9 + 12)-dimensional phase space,
·
which describes the interaction of the 8-dimensional isospin particle with S0(8)
monopole field. The dimensionality of the internal phase space of the particle is
equal to 12.
Now, we should proceed the last step: we need to modify the Lagrangian by
adding the specific term (vanishing for the lower Hopf maps), in such a way, that
not only I2, but each I will be the constant of motion of our system.
a
4 Conclusion
We have presented the reduction procedure associated with the first and second
Hopf map in the Lagrangian approach. For the last- the third Hopf fibration we
have presented the explicit formulae of the Lagrangian in coordinates of base and
fiber. Since we deal with irreducible representation of SO(8) algebra it is impossible
to construct the motion integrals corresponding to respective ones for the first and
second Hopf maps. The only way to avoid this problem seems to be modifying the
initial Lagrangian or considering non- Lie algebras of motion integrals.
Acknowledgments.
I am grateful to Armen Nersessian, Francesco Toppan, Zhanna Kuznetsova and
Marcelo Gonzales for collaboration on the work [1] which became the base for the
current one. Special thanks to Armen Nersessian for the useful discussions and
for the help in preparation of this paper. I would like to thank George Pogosyan
for given opportunity to give a talk at XIV International Conference on Symmetry
Methods in Physics(16-22 August, 2010), which hold In Tsakhkadzor, Armenia.
The work was partially supported by Volkswagen Foundation I/84 496 grants.
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